Question

Use the magnitudes​ (Richter scale) of the 120 earthquakes listed in the accompanying data table. Use technology to find the​ range, variance, and standard deviation. If another​ value,

8.00​, is added to those listed in the data​ set, do the measures of variation change​ much?

Magnitudes:

3.31
2.46
2.55
2.45
2.8
2.4
2.21
2.39
1.92
1.45
2.84
1.74
2.01
2.35
2.33
2.69
4.66
2.88
3.42
2.69
2.81
3.4
3.95
3.02
3.89
3.46
3.1
2.94
2.68
3.6
2.87
2.36
3.05
3.21
2.59
3.61
3.22
2.65
2.36
2.42
2.83
3.94
2.54
2.89
2.96
3.42
2.3
2.58
2.88
3.15
2.18
1.14
1.93
4.04
2.54
2.83
2.35
2.31
1.52
2.78
1.94
1.58
2.45
2.34
2.08
1.55
3.21
1.49
1.8
2.55
1.68
2.34
2.42
2.1
2.22
2.79
2.02
2.82
2.41
2.7
1.67
2.88
1.85
2
1.86
2.54
1.97
2.17
3.61
1.53
3.17
2.47
1.85
1.5
2.8
3.26
3.85
2.77
2.45
2.71
2.53
1.63
2.16
3.01
2.33
1.5
1.91
2.33
2.65
1.42
1.38
1.79
2.24
2.31
2.47
1.74
2.39
2.43
2.48
2.45

Answer 1

For the given data

The range is the difference between the highest and lowest values in the data set.

Ordering the data from least to greatest, we get:

1.14   1.38   1.42   1.45   1.49   1.5   1.5   1.52   1.53   1.55   1.58   1.63   1.67   1.68   1.74   1.74   1.79   1.8   1.85   1.85   1.86   1.91   1.92   1.93   1.94   1.97   2   2.01   2.02   2.08   2.1   2.16   2.17   2.18   2.21   2.22   2.24   2.3   2.31   2.31   2.33   2.33   2.33   2.34   2.34   2.35   2.35   2.36   2.36   2.39   2.39   2.4   2.41   2.42   2.42   2.43   2.45   2.45   2.45   2.45   2.46   2.47   2.47   2.48   2.53   2.54   2.54   2.54   2.55   2.55   2.58   2.59   2.65   2.65   2.68   2.69   2.69   2.7   2.71   2.77   2.78   2.79   2.8   2.8   2.81   2.82   2.83   2.83   2.84   2.87   2.88   2.88   2.88   2.89   2.94   2.96   3.01   3.02   3.05   3.1   3.15   3.17   3.21   3.21   3.22   3.26   3.31   3.4   3.42   3.42   3.46   3.6   3.61   3.61   3.85   3.89   3.94   3.95   4.04   4.66   

The lowest value is 1.14.

The highest value is 4.66.

The range = 4.66 - 1.14 = 3.52.

Now for variance we need to first calculate mean $\overline{x}=\frac{\sum x}{n}=2.52$

Create the following table.

data	data-mean	(data - mean)2
3.31	0.79	0.6241
2.46	-0.06	0.0036
2.55	0.03	0.0009
2.45	-0.07	0.0049
2.8	0.28	0.0784
2.4	-0.12	0.0144
2.21	-0.31	0.0961
2.39	-0.13	0.0169
1.92	-0.6	0.36
1.45	-1.07	1.1449
2.84	0.32	0.1024
1.74	-0.78	0.6084
2.01	-0.51	0.2601
2.35	-0.17	0.0289
2.33	-0.19	0.0361
2.69	0.17	0.0289
4.66	2.14	4.5796
2.88	0.36	0.1296
3.42	0.9	0.81
2.69	0.17	0.0289
2.81	0.29	0.0841
3.4	0.88	0.7744
3.95	1.43	2.0449
3.02	0.5	0.25
3.89	1.37	1.8769
3.46	0.94	0.8836
3.1	0.58	0.3364
2.94	0.42	0.1764
2.68	0.16	0.0256
3.6	1.08	1.1664
2.87	0.35	0.1225
2.36	-0.16	0.0256
3.05	0.53	0.2809
3.21	0.69	0.4761
2.59	0.07	0.0049
3.61	1.09	1.1881
3.22	0.7	0.49
2.65	0.13	0.0169
2.36	-0.16	0.0256
2.42	-0.1	0.01
2.83	0.31	0.0961
3.94	1.42	2.0164
2.54	0.02	0.0004
2.89	0.37	0.1369
2.96	0.44	0.1936
3.42	0.9	0.81
2.3	-0.22	0.0484
2.58	0.06	0.0036
2.88	0.36	0.1296
3.15	0.63	0.3969

$\sigma^2=\frac{\sum (xi-\overline{x})^2}{n-1}=0.4233$

So Standard deviation is $\sigma=\sqrt{0.4233}=0.6506$

Now adding 8 in the data we get

The range is the difference between the highest and lowest values in the data set.

Ordering the data from least to greatest, we get:

1.14   1.38   1.42   1.45   1.49   1.5   1.5   1.52   1.53   1.55   1.58   1.63   1.67   1.68   1.74   1.74   1.79   1.8   1.85   1.85   1.86   1.91   1.92   1.93   1.94   1.97   2   2.01   2.02   2.08   2.1   2.16   2.17   2.18   2.21   2.22   2.24   2.3   2.31   2.31   2.33   2.33   2.33   2.34   2.34   2.35   2.35   2.36   2.36   2.39   2.39   2.4   2.41   2.42   2.42   2.43   2.45   2.45   2.45   2.45   2.46   2.47   2.47   2.48   2.53   2.54   2.54   2.54   2.55   2.55   2.58   2.59   2.65   2.65   2.68   2.69   2.69   2.7   2.71   2.77   2.78   2.79   2.8   2.8   2.81   2.82   2.83   2.83   2.84   2.87   2.88   2.88   2.88   2.89   2.94   2.96   3.01   3.02   3.05   3.1   3.15   3.17   3.21   3.21   3.22   3.26   3.31   3.4   3.42   3.42   3.46   3.6   3.61   3.61   3.85   3.89   3.94   3.95   4.04   4.66   8.00   

The lowest value is 1.14.

The highest value is 8.00.

The range = 8.00 - 1.14 = 6.86.

Mean=$\overline{x}=\frac{\sum x}{n}=2.5653$

Create the following table.

data	data-mean	(data - mean)2
3.31	0.7447	0.55457809
2.46	-0.1053	0.01108809
2.55	-0.0153	0.00023409
2.45	-0.1153	0.01329409
2.8	0.2347	0.05508409
2.4	-0.1653	0.02732409
2.21	-0.3553	0.12623809
2.39	-0.1753	0.03073009
1.92	-0.6453	0.41641209
1.45	-1.1153	1.24389409
2.84	0.2747	0.07546009
1.74	-0.8253	0.68112009
2.01	-0.5553	0.30835809
2.35	-0.2153	0.04635409
2.33	-0.2353	0.05536609
2.69	0.1247	0.01555009
4.66	2.0947	4.38776809
2.88	0.3147	0.09903609
3.42	0.8547	0.73051209
2.69	0.1247	0.01555009
2.81	0.2447	0.05987809
3.4	0.8347	0.69672409
3.95	1.3847	1.91739409
3.02	0.4547	0.20675209
3.89	1.3247	1.75483009
3.46	0.8947	0.80048809
3.1	0.5347	0.28590409
2.94	0.3747	0.14040009
2.68	0.1147	0.01315609
3.6	1.0347	1.07060409
2.87	0.3047	0.09284209
2.36	-0.2053	0.04214809
3.05	0.4847	0.23493409
3.21	0.6447	0.41563809
2.59	0.0247	0.00061009
3.61	1.0447	1.09139809
3.22	0.6547	0.42863209
2.65	0.0847	0.00717409
2.36	-0.2053	0.04214809
2.42	-0.1453	0.02111209
2.83	0.2647	0.07006609
3.94	1.3747	1.88980009
2.54	-0.0253	0.00064009
2.89	0.3247	0.10543009
2.96	0.3947	0.15578809
3.42	0.8547	0.73051209
2.3	-0.2653	0.07038409
2.58	0.0147	0.00021609
2.88	0.3147	0.09903609
3.15	0.5847	0.34187409

$\sigma ^2=\frac{\sum (xi-\overline{x})^2}{n-1}=0.6679$

So $\sigma =\sqrt{0.6679}=0.8173$

Yes we see that values change